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07-036 Copyright © 2006, 2007, 2008 by Noel H. Watson Working papers are in draft form. This working paper is distributed for purposes of comment and discussion only. It may not be reproduced without permission of the copyright holder. Copies of working papers are available from the author. A Perceptions Framework for Categorizing Inventory Policies in Single-stage Inventory Systems Noel H. Watson

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  • 07-036

    Copyright 2006, 2007, 2008 by Noel H. Watson

    Working papers are in draft form. This working paper is distributed for purposes of comment and discussion only. It may not be reproduced without permission of the copyright holder. Copies of working papers are available from the author.

    A Perceptions Framework for Categorizing Inventory Policies in Single-stage Inventory Systems Noel H. Watson

  • A Perceptions Framework for Categorizing Inventory Policies inSingle-stage Inventory Systems

    Noel Watson

    Harvard Business School, Soldiers Field Road, Boston, MA 02163

    [email protected]

    July 31, 2008

    In this paper we propose a perceptions framework for categorizing a range of inventory decisionmaking that can be employed in a single-stage supply chain. We take the existence of a wide rangeof inventory decision making processes, as given and strive not to model the reasons that the rangepersists but seek a way to categorize them via their effects on inventory levels, orders placed giventhe demand faced by the inventory system. Using a perspective that we consider natural and thusappealing, the categorization involves the use of conceptual perceptions of demand to underpinthe link across three features of the inventory system: inventory levels, orders placed and actualdemand faced. The perceptions framework is based on forecasting with Auto-Regressive IntegratedMoving Average (ARIMA) time series models. The context in which we develop this perceptionsframework is of a single stage stochastic inventory system with periodic review, constant leadtimes,infinite supply, full backlogging, linear holding and penalty costs and no ordering costs. Forecast-ing ARIMA time series requires tracking forecast errors (interpolating) and using these forecasterrors and past demand realizations to predict future demand (extrapolating). So called optimalinventory policies are categorized here by perceptions of demand that align with reality. Naturallythen, deviations from optimal inventory policies are characterized by allowing the perception aboutdemand implied by the interpolations or extrapolations to be primarily different from the actualdemand process. Extrapolations and interpolations being separate activities, can in addition, implydiffering perceptions from each other and this can further categorize inventory decision making.

    1 Introduction

    In this paper we propose a perceptions framework for categorizing a range of inventory decision

    making in a single-stage supply chain. The perceptions framework is based on forecasting with

    Auto-Regressive Integrated Moving Average (ARIMA) time series models. In this paper, we take

    the existence of a wide range of inventory decision making processes as given and strive not to model

    the reasons that the range persists but seek a way to categorize them via their effects on inventory

    levels, orders placed given the demand faced by the inventory system. Using a perspective that we

    consider natural and thus appealing, the categorization involves the use of conceptual perceptions

    of demand to underpin the link across three features of the inventory system: inventory levels,

    orders placed and actual demand faced. With limited emphasis on the cost consequences of the

    1

  • inventory policies, (sake in the distinction of optimal and suboptimal,) our categorization is less an

    economics-based one, but rather process-based, focusing instead on the stochastic processes that

    characterize the inventory levels, orders and demand levels at the inventory system and how they

    can be considered inter-related. We also examine the implications of our inventory decision making

    categories with respect to the variance and uncertainty of these stochastic processes, characteristics

    which we expect would have significant influence on the costs for an inventory system and for its

    supply chain.

    Our use of the term perception is motivated by its use in organizational behavior theory on

    organizational decision making, March and Simon (1993), Cyert and March (1992). In such theory,

    perception of reality in terms of environment, decision alternatives, goals or consequences of actions

    are considered significant drivers of decision making. These perceptions can be misaligned with

    reality (assuming an objective assessment of this reality) since they are considered the outcome of

    psychological, sociological and operational processes with their attendant strengths and weaknesses.

    Why do we need such a framework? In the operations literature, the assumption of the com-

    pletely rational agent has dominated the research surrounding inventory policies. However devi-

    ations from the optimal policy are prevalent in practice and analytical models to understand the

    effects of inventory dynamics on practice may require means of modeling them. Modeling devi-

    ations from optimal policy can also augment our understanding of inventory systems and supply

    chains, in a manner akin to sensitivity analysis in mathematical programming, as the dynamics

    surrounding optimal management of these systems may not completely describe these systems. For

    example, although the optimal EOQ is determined via strict optimization of standard setup and

    holding costs, the inventory cost function is well-known to be fairly flat around this point.

    Inventory decision making, in addition to being influenced by the above mentioned perceptions

    of reality, will also be the outcome of a combination of psychological, sociological, organizational

    and operational processes, Oliva and Watson (2004, 2007). We refer to a particular combination

    of perceptions and decision making processes as an inventory decision making setting. In order to

    categorize these settings, we make the strong assumption that such settings can be grouped based

    on the inventory levels, orders and demand persistently exhibited within the system, despite these

    settings being otherwise considered different from each other. For example, when the operations

    literature has assumed a rational agent for particular inventory decisions say in response to i.i.d.

    demand, we would infer a reference to all settings where the particular combination of perceptions

    and decision making processes results in processes for inventory levels and orders along with de-

    mand that resemble rational behavior. In this paper, the commonality used for grouping settings

    2

  • is represented by stochastic processes, serving as conceptual perceptions of demand. These per-

    ceptions form both a conceptual and a satisfying analytic link between inventory levels and orders

    placed and the actual demand faced by the inventory. The details of this analytic link and thus

    the particular operationalization of our perceptions framework make use of the ARIMA time series

    models, Box et. al. (1994).

    From Box et. al., forecasting ARIMA time series requires tracking forecast errors (interpolating)

    and using these forecast errors and past demand realizations to predict future demand (extrapo-

    lating). Both interpolation and extrapolation are actions that can be considered to imply a belief

    about the demand process or a perception, as they are based on a particular time series demand

    model when performed by a completely rational agent. So called optimal inventory policies are

    categorized here by perceptions of demand that align with reality. Naturally then, categorizing

    deviation from optimal inventory policies is possible if we allow that the time series used for in-

    terpolating or extrapolating, that is, what we interpret as a belief about the demand process or

    perception, may not be the same as the actual demand process. Further deviations can even be

    categorized by allowing that the perception implied by interpolations do not match those implied

    by extrapolations.

    In this paper, we establish these conceptual perceptions as analytic links connecting inventory

    levels or orders and the actual demand. In so doing, we also provide a prediction for the effects

    of a range of inventory decision making settings on inventory levels and orders. For example, we

    can suggest the effects of inventory decision making that ignore any non-stationary properties of

    demand. In an effort to further our appreciation of the framework, we interpret simple forecasting

    techniques used for inventory policies such as the moving average and exponential smoothing using

    our framework. Interestingly we show that in order to categorize the use of the moving average, we

    need slightly different perceptions in interpolation versus that of extrapolation justifying the need

    for the frameworks flexibility. In an effort to improve our understanding of the effects of suboptimal

    policies on the process characteristics of inventory levels and orders, we also examine separately, the

    effects of misaligned perceptions along the autoregressive and then the moving average dimensions

    of the ARIMA modeled demand process and focus on the implications for variance and uncertainty

    of the inventory and order processes. Given that these are the two critical dimensions of the

    ARIMA processes which model our perceptions, understanding how each dimension separately

    affects inventory levels and orders should help our understanding of their effects in tandem.

    The use of ARIMA processes as the foundation for this perceptions framework is motivated by

    the following two important observations: First as cited in Gaur et. al. real-life demand patterns

    3

  • often follow higher-order autoregressive processes due to the presence of seasonality and business

    cycles. For example, the monthly demand for a seasonal item can be an AR(12) process. More

    general ARMA processes are found to fit demand for long lifecycle goods such as fuel, food products,

    machine tools, etc., as observed in Chopra and Meindl (2001) and Nahmias (1993). Second, the

    results in this paper along with recent research show that ARMA demand processes occur naturally

    in multistage supply chains. For example, Zhang (2004) studied the time-series characterization

    of the order process for a decision maker serving ARMA demand and using a periodic-review

    order-up-to policy. They show an ARMA-in-ARMA-out principle for perceptions aligned with

    actual demand, i.e., if demand follows an ARMA(p, q) process, then the order process is an ARMA

    (p,max [p, q l]) process, where l is the replenishment lead time. Gilbert (2005) extended these

    results to ARIMA process but still for aligned perceptions. We find a similar ARIMA-in-ARIMA-

    out principle for orders given a wider range of inventory policies. Such a result increases our

    expectations for the prevalence of these processes in real world supply chains.

    It needs to be pointed out that in our framework we do not use perception in its more con-

    ventional sense. We are not making any claims about the actual perception of any manager in

    this paper and how decisions are made by these managers. However, there are real world scenar-

    ios which resemble details of our perception framework more closely than others. One scenario

    involves fitting ARIMA stochastic models to historical demand data, Gooijer (1985), Koreisha

    and Yoshimoto (1991). Different methods for fitting these stochastic processes including the Box-

    Jenkins approach, the corner method and extended sample autocorrelations, have been shown to

    have varying performance in correctly identifying these processes. Poor identification then could

    result in a situation where the demand process used for inventory management is misaligned with

    reality. A second scenario involves subjective forecasting. Lawrence and OConnor (1992) in a fore-

    casting study investigate the extent to which some of the widely documented judgemental biases

    and heuristics apply to time series forecasting. The time series used were modeled from stationary

    ARMA processes. The authors found that subjects forecasts could be modeled as an AR(1) process

    that was not always aligned with actual.

    Supply chain issues which we expect could benefit from our framework include the bullwhip

    effect and its mitigation Lee, et. al. (1997), value of information sharing in a supply chain, Lee, et.

    al. (2000), Cachon and Fisher (2000), Raghunathan (2001), Gaur et. al. (2005) and the general

    coordination of a supply chain whether via incentives Cachon (2000) or more structural approaches

    such as leadtime reduction, Cachon and Fisher, or network rationalization. For example, in section

    4.2, we make some comments on the value of sharing information in a supply chain. In Watson and

    4

  • Zheng (2005), misaligned perceptions implying deviations from optimal inventory policies, were

    also used to test the robustness of decentralization schemes for serial supply chains, that is, how

    well the schemes mitigated the resulting increase in total systems costs.

    Section 2 presents the general demand model and inventory system while Section 3 introduces

    the perceptions framework within the context of optimal inventory management. Section 4 con-

    tinues the introduction but within the context of suboptimal inventory policies focusing, for sake

    of simplicity, on true demand modeled as stationary ARMA stochastic processes. Section 5 exam-

    ines misaligned perceptions along the autoregressive and moving average dimensions separately. In

    Section 6 we also examine policies based on simple forecasting techniques. Section 7 completes the

    presentation of the perceptions framework within the context of suboptimal inventory policies with

    a treatment of demand modeled as non-stationary ARIMA processes.

    2 The Demand Model and Inventory System

    In this paper, we use the general class of Auto-Regressive Integrated Moving Average (ARIMA)

    time-series models to model both demand and perceptions of demand. In this section we describe

    the methodology for forecasting with these time series models once they have been specified. Since

    perceptions are also modeled as ARIMA time series models, the description is applicable for un-

    derstanding the details of the mechanics of the perceptions framework.

    We follow the notation of Box et. al. (1994). Stationary demand will be represented by an

    autoregressive moving average model, ARMA(p, q) with p the number of autoregressive terms and

    q the number of moving average terms:

    Zt = + 1 (Zt1 ) + 2 (Zt2 ) + ...+ p (Ztp ) + at 1at1 2at2 ... qatq

    (1)

    = +

    pXi=1

    i (Zti ) + at +qXi=1

    iati,

    where Zt is the demand at time, is the level or mean of demand and at is a noise series of

    independent identically distributed random variables with mean 0 and variance 2a.

    The ARMA models can be represented more concisely and are more easily manipulated math-

    ematically by using a backshift operator. Let B denote the time-series backshift operator such

    that

    BZt = Zt1 and BnZt = Ztn.

    5

  • The above stationary series (1) can be written using the backshift operator as follows:

    (B) (Zt ) = (B) at,

    where (B) = 1 1B 2B2 ... pBp and is referred to as the autoregressive operator with

    order p, and (B) = 11B2B2 ...qBq which is referred to as the moving average operator

    with order q.

    Non-stationary demand can be modeled by assuming that some differencing results in a station-

    ary ARMA series. will be used to denote the difference operation such that Zt = (1B)Ztand dZt = (1B)d Zt where d = (1B)d is the polynomial resulting from raising 1B to the

    dth power. Assuming the dth difference of a non-stationary demand series is stationary, the series

    can be represented by an autoregressive integrated moving average or ARIMA(p, d, q) series:

    (B)dZt = (B) at (2)

    or

    (B)Zt = (B) at,

    where (B) = (B)d is a polynomial of order p+ d and is referred to as the generalized autore-

    gressive operator .

    2.1 ARIMA(p, d, q) Forms and Forecasting

    There are three different explicit forms for the general model (2) . The current value of Zt can

    be expressed:

    1. In terms of the realizations of previous Ztj : j > 0 and current and previous realizations

    atj , j = 0, 1, ... as implied by (2) :

    (Zt ) = (1 (B)) (Zt1 ) + (B) at.

    2. In terms of the current and previous realizations atj , j = 0, 1, .... This representation is seen

    by rewriting (2) as follows

    (Zt ) = (B) at, (3)

    where (B) = 1 +Pj=1 jB

    j = (B) / (B) . In particular:

    j = p+djpd + ...+ 1j1 j : j > 0

    6

  • where 0 = 1, j = 0 for j < 0, and j = 0 for j > q. For the ARMA time series, the stationary

    property is guaranteed if the series (B) is infinitely summable, that is, ifPj=1 |j | < .

    For an ARIMA time series Zt, although dZt is stationary, Zt is not. Therefore the infinite

    sum (3) above, is not summable and so does not converge in any sense. Alternatively we

    can rewrite (18) in terms of the dth difference which does have a converging sum of random

    shocks. As a notational device, for ARIMA time series we will refer to this infinite random

    shock form to represent the infinite random shock form of the dth difference.

    3. In terms of the realizations of previous Ztj and current at. This representation is seen by

    rewriting (2) as follows

    (B) (Zt ) = at,

    where (B) = 1Pj=1 jB

    j = (B) / (B) . This implies the following

    (Zt ) =Xj=1

    j (Ztj ) + at.

    In particular:

    j = qjq + ...+ 1j1 + j : j > 0

    where 0 = 1, j = 0 for j < 0 and j = 0 for j > p. An invertibility property for (2) implies

    that this is a legitimate form. The property is guaranteed by restricting the series (B) to

    be infinitely summable, that is,Pj=1 |j |

  • on-hand inventory, and then 3) orders are placed. Let It be the inventory at the retailer at the end

    of period t. Consistent with the assumptions on the costs of the inventory system, we will assume

    an order up-to policy for the retailer with target st for period t.Where necessary we allow costless

    returns which implies that orders from the retailer in period t are given by

    Ot = Zt + st+1 st. (4)

    The primary purpose of this paper is to examine the use of perceptions as a framework for

    categorizing inventory policies, therefore we will for simplicity and brevity assume that the inventory

    system has zero leadtime, that is, L = 0. The results for arbitrary leadtime have similar themes.

    3 The Perceptions Framework Applied to Optimal Inventory Poli-cies

    In this section we introduce the perceptions framework within the context of optimal inventory

    management.

    3.1 Aligned Perceptions: Optimal Inventory policies.

    Consider the general demand process for the retailer (2) and the optimal inventory policy for

    minimizing expected single period costs which, in this setting, is the myopic order up-to policy.

    For this optimal policy, at the end of period t 1, the retailer calculates the forecast Zt1 which

    minimizes the forecast error for period t and combines this forecast with a constant safety stock

    T . From Box et. al. as mentioned earlier this forecast can be generated using any of the three

    explicit forms of the demand process. The first two forms require, however, the retailer to calculate

    at least the most recent random shock at1. This is calculated as the difference between the actual

    realization and the one period ahead forecast, that is, at1 = Zt1 Zt2 for an ARMA time

    series or at1 = dZt1 dZt2 for an ARIMA time series. (Here dZt2 serves as convenient

    notation for the forecasted dth difference.) When our retailer calculates the difference between the

    actual realization and the one period ahead forecast, she interpolates the realizations of the unseen

    random shocks. When the retailer makes her forecast Zt1 using any of the explicit forms, she is

    extrapolating, that is, using previous realizations and interpolations to predict future realizations.

    Both interpolation and extrapolation as actions can be considered to imply a belief about the

    demand process as they are based on a particular time series model. We attribute this time series

    implied by either forecasting operation of interpolation or extrapolation as a perception about the

    demand process. The resulting optimal forecasts and inventory policies can then be considered

    8

  • optimal partly because the perception of the retailer, that is, the belief of the demand process,

    is aligned (or matches) with the actual demand process. The terms introduced here, perception,

    interpolation and extrapolation will be used in the next section for categorizing a wider range of

    inventory policies beyond the optimal policy for an inventory system.

    The theorems below characterize the stochastic process for the inventory levels at the retailer

    and her orders assuming the myopic policy described above and comes from Gilbert. These results

    will be compared with those obtained when we examine the perceptions framework within a context

    of misaligned perceptions of demand.

    Theorem 3.1 An aligned perception is associated with a time series of inventory It which is

    ARIMA(0,0,1) with the form It T = aIt where aIt = at.

    Theorem 3.2 An aligned perception is associated with a time series of orders Ot which is

    ARIMAp, d, qO

    with the form

    (B) (Ot ) = O (B) aOt

    where aO = KOat, O (B) =(B)(B)BKO

    + (B)KO

    , qO = max (p+ d, q 1) and KO = 1 + 1 1.

    Proposition 3.3 For stationary ARMA demand process, the variance of the time series (Ot )

    is greater than that of (Zt ) if and only if 1 1 > 0.

    3.2 Discussion of Optimal Policy

    Theorem 3.1 suggests that inventory would rarely follow a process more complex than a simple

    i.i.d. process despite the complexity of the demand process and further, that the mean of this

    inventory process is equal to the intended safety stock T. Theorem 3.2 shows that the optimal

    orders follow an ARIMA process similar to that of demand except for differences in the variance of

    the random shock and in the moving average operator of the process. The random shock for the

    order process has varianceKOa

    2= (1 + 1 1)2 2a. In Gilbert and Watson and Zheng (2008)

    the coefficient |KO| is interpreted as an alternative measure of the bullwhip effect complementing

    the more conventional measure of the ratio of variance of orders to demand. Watson and Zheng

    refer to the coefficient |KO| as the bullwhip effect modulo full information and interpret it as the

    best case increase (or decrease) in uncertainty for a potential supplier given full distributional

    knowledge about demand and knowledge about the inventory policy of the retailer. The greater

    the coefficient |KO|, the greater the expected uncertainty for the supplier despite her extensive

    9

  • information. Proposition 3.3 shows that for aligned perceptions, this bullwhip effect modulo full

    information is consistent with the traditional measure of the bullwhip effect.

    The results for aligned perceptions here provide added argument for the need for a framework on

    deviations from the optimal policy. Theorem 3.1 provides very strict expectations for the inventories

    of a retailer, (that it follows a ARIMA (0,0,1) process,) which we would be surprised, if we found

    completely characterized a majority of inventory systems.

    3.3 Formalizing Framework Concepts: Perceptions, Interpolations and Extrap-olations

    We assimilate the concepts of perception, interpolation and extrapolation introduced in the previous

    subsections within a context of forecasting with ARIMA time series models in order to provide

    our framework. Perception we will define as the implied characterization of the demand process

    facing the inventory system. Conceptually, perceptions can be implied from interpolations and

    extrapolations. In this paper, interpolation is the action of inferring past but hidden realizations,

    in this case, the unseen random shock from demand realizations. Extrapolation is the action

    of using historical realizations and/or interpolations to predict future demand realizations. In

    order to impute a characterization of demand, that is, a perception, from an action requires an

    explicit mapping of demand processes to actions. Our mapping corresponds to that generated by a

    completely rational agent making forecasts for a given ARIMA demand specification. In essence, a

    perception is implied from an interpolation or extrapolation by assuming the action to be performed

    by a completely rational agent and then inferring the demand process which drives the action. By

    our definition of perception, it will not be necessary for the perception imputed from interpolations

    to be the same with that imputed from extrapolations (as implicitly assumed in the optimal policy

    above) even though the implicit assumption is that the actions (and perceptions) categorize the

    same inventory system. More importantly neither of these perceptions need be aligned with reality.

    Using the above introduced definitions, we can examine a range of inventory policy responses by

    our retailer and study the effects on her orders and inventory levels.

    4 Perceptions Framework for Suboptimal policies: Misaligned Sta-tionary Perceptions

    In this section we move beyond optimal policies and begin to examine inventory decision mak-

    ing that can be termed suboptimal. In our framework, such policies are categorized by percep-

    tions which do not align with reality and thus we will term them misaligned perceptions. In this

    10

  • section we will concentrate on stationary demand processes and stationary perceptions, that is,

    ARMA (ARIMA with d = 0) processes as in (1) , while in Section 7 we focus on non-stationary

    ARIMA (d > 0) demand processes. Our categorization will be based on two perceptions, one for

    interpolations and one for extrapolations. Therefore we can consider two variations in the extent

    of the misaligned perceptions. The first (section 4.2) considers the same misaligned perceptions in

    both extrapolations and interpolations and the second considers different perceptions for extrapola-

    tions and interpolations. Although the first variation concerns perceptions that could be described

    as internally consistent since that used for extrapolation matches that used for interpolations, the

    second variation follows from recognizing that these perceptions do not have to be so. In section

    4.3 we consider one example where the perception used for interpolation is different from that used

    for extrapolation. In particular we consider misaligned perceptions in extrapolations but aligned

    perceptions in interpolations. Furthermore, in section 6, we interpret the use of the moving av-

    erage and exponential smoothing using our perceptions framework. The example of the moving

    average policy considered in section 6.1 will serve as another example where the perception used

    for interpolation will be different from the perception used for extrapolation. Our main goal in this

    section (and paper) is to establish these perceptions as the fundamental component of an analytic

    link between the demand process and the process for inventory levels and orders. In so doing, we

    also provide a prediction for the effects of suboptimal inventory decision making on inventory

    levels and orders.

    4.1 Misaligned Stationary Perceptions : ARMA(p, q) processes

    In this subsection we discuss the general stationary perceptions that can be attributed to inter-

    polations and extrapolations which we will allow to differ from the actual demand process. All

    stationary and invertible ARMA processes can serve as misaligned perceptions, thus follow an

    ARMA(p, q) process:

    (B) (Zt ) = (B) at (5)

    where (B) = 1 1B 2B2 ... pBp, (B) = 1 1B 2B2 ... qBq, and at is a random

    noise series with mean 0 and variance a. The stationary and invertible properties from section 2.1

    are assumed to hold and thus all three explicit forms of forecasting will be legitimate. In (5) it is

    possible for p = p, q = q, = , j = j , j = j for some j or a = a.

    We will assume that the forecasts for extrapolations are denoted by Zet [k] which specifically

    denotes the forecast of demand at time t + k made at time t after demand is observed. Since we

    concentrate on zero leadtimes, we only deal with single period ahead forecasts, Zet [1] , which we

    11

  • will denote Zet . Recall from section 2.1 that there are three different explicit forms for the general

    ARIMA models and that these forms imply three different approaches to calculating Zet . We will

    denote these three forms as Zet (1), Zet (2) and Z

    et (3) which coincide sequentially with the three

    explicit forms from Section 2.1 and define them explicitly as follows:

    Zet (1) = +1 (B)

    B(Zt ) +

    (B) 1B

    at, (6)

    Zet (2) = + (B) 1

    Bat, (7)

    Zet (3) = +1 (B)

    B(Zt ) , (8)

    where (B) = (B) / (B) and (B) = (B) / (B) . The above formulas are generated by taking

    the expectation at time t of demand in period t+ 1, Zt+1, using the three explicit forms listed in

    Section 2.1 for the process (5). Note that at time t, Eat+1 = 0. Furthermore the realizations ak,

    k t, are determined from interpolations. Note finally that Zet (3) does not use any interpolations.

    For interpolations, recall that it is the difference between the actual realization and the one

    period ahead forecast that is used to generate the realization for the demand shock at that are used

    for extrapolations in (6) and (7) above. We let Zit denote this one period ahead forecast based

    on the perceptions used for the interpolation. As with extrapolation, we can denote three forms

    for Zit as Zit (1), Z

    it (2) and Z

    it (3) which again coincide sequentially with the three explicit forms

    listed in Section 2.1. Specifically at = (Zt ) Zit1 (J)

    where the forecast used in the

    interpolation could be of any of the three forms J = 1, 2, 3. For J = 3, is it is easy to show that we

    can then write at = (B) (Zt ) where (B) = (B) / (B) . Note that since the perception (5)

    is invertible, (B) is legitimate. Note that the other two forms J = 1, 2 also make use of demand

    realizations determined from prior interpolations. For these forms it only takes a little more work

    to show that we still get at = (B) (Zt ) . We write this as a proposition:

    Proposition 4.1 For misaligned interpolations, either form of the single period ahead forecast

    Zit (J) , J = 1, 2, 3, results in realization of demand shocks which can be formulated as at =

    (B) (Zt ) where (B) = (B) / (B) .

    4.2 Consistent Misaligned Perceptions in Interpolation and Extrapolation

    In this subsection by characterizing the stochastic processes for the inventory levels and orders

    given the same misaligned perception in both interpolations and extrapolations, we establish this

    perception as the basis for the link between the process characteristics of the system. After the

    12

  • results provided below we describe the more interesting relationships between the categorizing

    perception and the process results.

    Assume without loss of generality that this misaligned perception is defined as the ARMA(p, q)

    process (5) . We have the following theorems:

    Theorem 4.2 A misaligned perception attributed to both interpolation and extrapolation is associ-

    ated with a time series of inventory It which is ARMApI , qI

    is with the form

    I (B)It T I

    = I (B) aIt

    and:aIt = K

    Iat pI = q + p I (B) = (B)(B)

    KI

    KI = 1 qI = p+ qT I = T +

    1B(B)

    B

    ( ) I (B) = (B) (B)

    This time series is stationary and independent of the explicit forecasting form used for either

    interpolation or extrapolation. The time series is also invertible if the demand process is invertible.

    Theorem 4.3 A misaligned perception attributed to both interpolation and extrapolation is associ-

    ated with a time series of orders Ot which is ARMApO, qO

    with the form

    O (B)Ot

    = O (B) aOt

    and:aOt = K

    Oat pO = p+ q O (B) =

    (B)(B)(1B)

    BKO

    (B)

    KO = 1 + 1 1 qO = max (p+ q, q + q 1)O (B) = (B) (B)

    This time series is stationary and independent of the explicit form used for either interpolation

    or extrapolation but not necessarily invertible.

    Perceptions Perspective on the Inventory Level Process: Theorem 4.2 above shows that

    the inventory level process for a misaligned perception in both interpolations and extrapolations

    is not i.i.d. (versus results for aligned perception in Theorem 3.1) but can follow a wide range of

    ARMA processes. The inventory level process is driven by both the perception and the demand

    process. In particular, the autoregressive and moving average operators from both the perception

    and the demand process determine the operators of the inventory level process. Furthermore, since

    T I = T +1B(B)

    B

    ( ) , a perception which has mean 6= results in essentially a different

    safety factor as the mean of the inventory level process. Here notice that except for the mean of

    actual demand, this safety factor is completely determined by the perception and is independent

    13

  • of the stochastic process for actual demand. As expected this safety factor T I is usually lower

    than T if since1B(B)

    B

    tends to be negative (since -1 is the constant in the polynomial

    1B(B)B

    ,) however this relation does not have to hold.

    Perceptions Perspective on the Order Process: As with the inventory level process,

    the order process is determined by the perception and the demand process. This is an important

    observation given its relationship to supply chain research on sharing demand for improving supply

    chain coordination, e.g., Lee, So and Tang. Researchers have argued that sharing such demand

    would help supply chain coordination in three ways: 1) historical demand information would help

    upstream stages forecast end-customer demand more accurately, which would then improve the

    forecasts of future orders from downstream stages, 2) knowing real-time demand realizations would

    also improve accuracy of forecasts of future orders, Lee, So and Tang, and 3) end-customer demand

    can prove a more robust guide for orders placed by upstream partners than their incoming orders,

    Watson and Zheng (2005). Our perceptions perspective here speak to the first two arguments. To

    the first argument, our results suggest that for improving forecasts of orders, an understanding of

    the demand process needs to be complemented by an understanding of the inventory policy of the

    downstream stage creating the orders. Historically, these orders have been assumed to come from

    aligned perceptions with the demand process, but if in reality these perceptions are not aligned,

    then forecasting demand is not sufficient for forecasting incoming orders. In particular, we see in

    Theorem 4.3 that the bullwhip effect modulo information for orders |KO|, that is the uncertainty

    inherent in orders, is related only to the parameters of the misaligned perception and not to that of

    the actual demand process. With respect to the second argument, the results suggest interestingly

    that sharing current demand realizations does not improve accuracy of order forecasts since the

    order process is also ARMA and thus completely characterized by previous orders and interpolated

    errors in forecasting, see Raghunathan (2001) for similar finding with aligned perceptions.

    4.3 Inconsistent Misaligned Perceptions in Extrapolation and Interpolation

    In this subsection we continue our examination of inventory decision making that can be termed

    suboptimal, focusing on those which are categorized by differing perceptions in extrapolations

    versus interpolations. We will consider a simple example of this variation where the perceptions

    from extrapolations are not aligned with the actual demand process (1), while the perceptions

    from interpolations are so aligned. This variation is considered to be one example but not the

    only such example where the perception used for interpolation is different from that used for

    extrapolation. Our results show that allowing perceptions to differ between interpolations and

    14

  • extrapolations, allows us to categorize a larger range of inventory decision making settings than

    when the perceptions are not allowed to differ. The example of the moving average policy considered

    in section 6.1 will serve as an example where the perception used for interpolation will be different

    from the perception used for extrapolation.

    First consider interpolations. Since the perceptions here is of a belief that matches with reality,

    we assume that the one period ahead forecasts used here, Zit , are equal to those of the completely

    rational agent, that is, Zit = Zt. This gives generated realizations for the random shocks, which will

    be used in extrapolation, that match with reality, i.e., at = at.

    Now consider extrapolations. Assume w.l.o.g. that the misaligned perception attributed to our

    extrapolation follows the ARMA(p, q) process (5) . For clarity, we write below the three explicit

    forms of the one period ahead forecasts:

    Zet (1) = +1 (B)

    B(Zt ) +

    (B) 1B

    at; (9)

    Zet (2) = + (B) 1

    Bat; (10)

    Zet (3) = +1 (B)

    B(Zt ) ; (11)

    Note that in the first two explicit forms of Zet we use the random shock terms at instead of at to

    reflect interpolations with aligned perceptions.

    The following are theorems characterizing the inventory levels and orders from our retailer for

    the first and second explicit forecasting form. Note that since the third explicit forecasting form

    is the same here as when extrapolations and interpolations are both misaligned, the results in

    theorems 4.2 and 4.3 apply in that case. The first difference to note between the results here and

    in section 4.2 is that here for inconsistent misaligned perceptions, the form of forecasting affects

    the results for inventory and orders, whereas for consistent misaligned perceptions in section 4.2,

    it does not.

    Theorem 4.4 A misaligned perception in extrapolation only and where extrapolations use the first

    explicit form of forecasts, is associated with a time series of inventory It which is ARMAp, qI

    with the form

    (B) (It T I) = I (B) aIt

    and:aIt = K

    Iat qI = max {p+ q 1, p+ q, p+ q}

    KI = (1 + 1 1) I (B) = 1KI(B)(B)

    B (B) + (B) (B) 1 (B)

    T I = T +

    1B(B)

    B

    ( )

    15

  • and associated with a time series of orders Ot which is ARMAp, qO

    with the form

    (B)Ot

    = O (B) aOt

    and:aOt = K

    Oat qO = max {p+ q, p+ q}

    KO = 1 + 1 1 O (B) = 1KO(B)B

    (B)

    (B) 1

    (B) (B)

    (1B)B

    Both time series It and Ot are stationary but not necessarily invertible.

    Theorem 4.5 A misaligned perception in extrapolations only and where extrapolations use the

    second explicit form of forecasts, is associated with a time series of inventory It which is ARMApI , qI

    with the form

    I (B) (It T I) = I (B) aIt

    and:aIt = K

    Iat qI = max {p+ q, q + p, p+ p}

    KI = 1 I (B) = 1KI

    (B) (B) (B) (B) (B) (B)

    T I = T ( )pI = p+ p I (B) = (B) (B)

    and associated with a time series of orders Ot which is ARMApO, qO

    with the form

    O (B)Ot

    = O (B) aOt

    and:aOt = K

    Oat qO = max {p+ q, q + p, p+ p}

    KO = 1 + 1 1 O (B) = 1KO (B) (B) +(B)(B)

    B (B) (1B)pO = p+ p O (B) = (B) (B)

    Both time series It and Ot are stationary but not necessarily invertible.

    Theorems 4.4 and 4.5 above show that the resulting processes for inventory and orders depend

    on which form of forecasting is used. For example, inventory and orders under the first form of fore-

    casting in Theorem 4.4 have the same autoregressive operator as the demand process while under

    the second form, in Theorem 4.5, the autoregressive operator for inventory and orders can be quite

    different from actual demand, i.e., (B) 6= I (B) and (B) 6= O (B). Comparing Theorems 4.4

    and 4.5 for inconsistent misaligned perceptions with Theorems 4.2 and 4.3 for consistent misaligned

    perceptions further shows how the inventory and order processes can differ when the perceptions

    are not consistent. For example, the variational coefficient KI in the inventory model in Theorem

    4.4 is rarely equal to -1 for inconsistent misaligned perceptions while it is always equal to -1 in The-

    orem 4.2. Furthermore inconsistent misaligned perceptions can result in inventory level processes

    16

  • that are not invertible while consistent misaligned perceptions always result in invertible inventory

    level processes if the demand process is invertible. This implies that allowing the perceptions for

    interpolations and extrapolations to differ from each other can help categorize a greater range of

    inventory decision making settings than when the perceptions are confined to be the same.

    5 Investigating the Effects of Consistent Misaligned Perceptionsalong Auto-Regressive and Moving Average Dimensions

    In an effort to improve our understanding of the effects of suboptimal policies on the process

    characteristics of inventory levels and orders, and promote the utility of our perceptions framework,

    we examine perceptions where the misalignment is either along the autoregressive dimension or

    moving average dimension but not both. Given that these are the two critical dimensions of the

    ARIMA processes which model our perceptions, understanding how each dimension separately

    affects inventory levels and orders should help our understanding of their effects in tandem. We

    will refer to these misalignments as autoregressive based misalignments and moving average based

    misalignments respectively and restrict our attention to consistent misaligned perceptions. We will

    examine the implications of both types of misalignment with respect to the variance and uncertainty

    of the inventory level and order processes.

    5.1 Auto-regressive based Misalignments

    In this subsection we consider misaligned perceptions with the misalignment along the autore-

    gressive dimension. Given that demand follows the ARMA process (1), an autoregressive based

    misaligned perception is the following ARMA(p, q) process

    (B) (Zt ) = (B) at (12)

    where (B) = 1 1B 2B2 ... pBp. Note that we assume that the mean of perception is

    the same as that of actual demand. The following proposition will be used to discuss this type of

    misalignment.

    Proposition 5.1 An autoregressive based misaligned perception in both the interpolation and ex-

    trapolation is associated with a time series of inventory It which is ARMAp, qI

    is with the form

    (B) (It T ) = I (B) aIt

    and:

    17

  • aIt = KIat K

    I = 1 qI = p I (B) = (B)

    KI

    and associated with a time series of orders Ot which is ARMA

    p, qO

    with the form

    (B) (Ot ) = O (B) aOt

    and:

    aOt = KOat q

    O = max (p, q 1) KO = 1 + 1 1 O (B) = (B)BKO (B)(1B)BKO

    These inventory and order time series are independent of the explicit form used for either

    interpolations or extrapolations.

    Perception Perspective on the Inventory Level Process: Comparing results for aligned

    perceptions from Theorem 3.1 with results for autoregressive based misalignments from Proposition

    5.1, we see that an autoregressive based misalignment introduces an autoregressive operator into the

    inventory level process which is the same as that of the demand process. Recall that under aligned

    perceptions, the inventory level process was i.i.d. A moving average operator is also introduced

    into the inventory level process. As expected, in Proposition 5.2, we show that aligned perceptions

    result in the lowest variance for the inventory process.

    Proposition 5.2 If (B) 6= (B) thenIt T

    has a larger variance than (It T ) .

    Monotonic results for changes in variance of inventory with changes in parameters of the mis-

    aligned perception depend on the parameters of actual demand. In Proposition 5.3 below when the

    coefficients of the autoregressive operator for actual demand are all positive and either all greater

    than or less than those of the misaligned perception, then variance will increase with the divergence

    of the misaligned perception from that of actual demand. Such monotonic properties do not always

    hold however for example, if some of the autoregressive coefficients are negative.

    Proposition 5.3 Assume p p and i 0, for all i, if i i 0 for all i or i i 0 for all

    i, then the variance of It is increasing ini i

    for all i.

    Perception Perspective on the Order Process: Comparing results for aligned perceptions

    from Theorem 3.2 with results for autoregressive based misalignment from Proposition 5.1, we see

    an autoregressive based misalignment does not change the autoregressive operator of the order

    process from what would have resulted from an aligned perception. Furthermore, the bullwhip

    effect modulo full information, that is the uncertainty inherent in orders, is determined by the

    18

  • mis-estimate of the first coefficient of the autoregressive operator 1. When 1 1, this implies

    that the bullwhip effect modulo full information is greater than for aligned perceptions, else lower.

    With respect to the variance of orders, Proposition 5.4 provides monotonic results for changes in the

    coefficients of the autoregressive operator of the perception. We find that given positive coefficients

    in the autoregressive operator of demand, the variance of orders increases as the coefficients of the

    autoregressive operator of the perception increase. The assumption in the proposition that i for

    i 1 is a decreasing sequence is not too onerous due to the assumption of the stationarity property

    but it does mean that i 0 for all i.

    Proposition 5.4 Given Ot = (B) aot , assume that the sequence i for i 1 is a decreasing

    sequence. If i 0 for all i, then for all i, V arOt

    increases as i increases.

    5.2 Moving Average based Misalignments

    In this subsection we consider misaligned perceptions with the misalignment along the moving

    average dimension. Given that demand follows the ARMA process (1), a moving average based

    misaligned perception is the following ARMA(p, q) process

    (B) (Zt ) = (B) at. (13)

    where (B) = 1 1B 2B2 ... qBq. Note that we again assume that the mean of perception

    is the same as that of actual demand.

    Proposition 5.5 A moving average based misalignment in both the interpolation and extrapolation

    is associated with a time series of inventory It which is ARMApI , q

    is with the form

    I (B) (It T ) = (B) aIt

    and:aIt = K

    Iat pI = q I (B) = 1

    KI

    (B)

    KI = 1and associated with a time series of orders Ot which is ARMA

    pO, qO

    with the form

    O (B) (Ot ) = O (B) aOt

    and:aOt = K

    Oat pO = q + p O (B) = (B) (B)

    KO = 1 + 1 1 qO = max (p+ q, q + q 1) O (B) =

    (B)BKO

    (B)(1B)BKO

    (B)

    This order time series is independent of the explicit form used for either interpolations or

    extrapolations.

    19

  • Perceptions Perspective on the Inventory Level Process: Comparing the results for

    aligned perceptions in Theorem 3.1 with the results for moving average based misalignments in

    Proposition 5.5, we see that a moving average based misalignment introduces a moving average

    operator to the inventory process which is the same as that of the demand process. Recall again

    that under aligned perceptions, the inventory level process was i.i.d. An autoregressive operator

    is also introduced into the inventory process since pI = q. As expected, from Proposition 5.6

    below, misaligned perceptions increase the variance of the inventory level process versus aligned

    perceptions.

    Proposition 5.6 If (B) 6= (B) thenIt T

    has a larger variance than (It T ) .

    As with autoregressive based misalignments, monotonic results for changes in variance of inven-

    tory with changes in parameters of the misaligned perception depend on the parameters of actual

    demand. In Proposition 5.7 below, when the coefficients of the moving average operator of the per-

    ception are all positive and either all greater than or less than those of the demand, then variance

    will increase with the divergence of the misaligned perception from that of actual demand. Such

    monotonic properties do not always hold however for example, if some of all of the moving average

    coefficients in the misaligned perception are negative.

    Proposition 5.7 Assume q q and i 0, for all i, if i i 0 for all i or i i 0 for all

    i, then the variance of It is increasing ini i

    for all i.

    Perception Perspective on the Order Process: Comparing the results for aligned percep-

    tions in Theorem 3.2 with the results for moving average based misalignments in Proposition 5.5,

    we see that a moving average based misalignment has an effect on both the autoregressive and on

    the moving average dimensions of the order process compared to aligned perceptions. The bull-

    whip effect modulo full information is determined by the mis-estimate of the first moving average

    coefficient 1. If 1 1 then the uncertainty surrounding orders is less than that for aligned per-

    ceptions, else greater. With respect to the variance of orders we are unable to prove monotonicity

    results for changes in the parameters of the misaligned perception. Rather, Proposition 5.8 pro-

    vides monotonic results for changes in the coefficients of the moving average operator of the actual

    demand holding the moving average operator of the misaligned perception constant. (This is the

    reverse of our findings for the autoregressive based alignments above, justifying somewhat the need

    for separating the analysis along the two dimensions as we do here.) We find that given positive

    coefficients in the moving average operator of perception, the variance of orders increases as the

    20

  • coefficients of the moving average operator of the demand increase. Again the assumption that

    i for i 1 is a decreasing sequence is not too onerous due to the assumption of the stationarity

    property but it does mean that i 0 for all i.

    Proposition 5.8 Given Ot = (B) aot , assume that the sequence i for i 1 is a decreasing

    sequence. If i 0 for all i, then for all i, Ot increases as i increases.

    6 Perceptions Perspective on Moving Average Forecasting andExponential Smoothing

    In this section we will see how simple forecasting approaches such as the moving average and

    exponential smoothing can be examined using the perceptions framework. The main goal will be

    to describe the perceptions that could be attributed to the use of these forecasting approaches. For

    completeness, the predictions on the stochastic processes for inventory levels and orders are also

    provided.

    6.1 Moving Average Forecasting

    The moving average involves the use of the simple average of the previous n demand realizations,

    Znt as an estimate of the mean of the demand process. Here

    Znt =

    Pni=1 Zti+1n

    .

    Forecast errors for the moving average are calculated as Znti Zti+1 for all i and the simple

    average of the previous n forecast errors, nt is also used as an estimate of the standard deviation of

    the forecast error and used to determine a proxy for the safety stock of the inventory policy. Here

    nt =

    Pni=1

    Znti Zti+1

    n

    . (14)

    The order up-to target for the inventory system is Znt + nt k with

    nt k as the implied safety factor.

    Here the estimate of the mean of the demand process and the safety factor comprise the order-up-to

    target, implying that our safety stock T = 0. This results in the following claim for inconsistent

    misaligned perceptions which categorize the use of the moving average for setting inventory policies.

    The claim follows since it directly mimics the use of the moving average just described and lemmas

    6.1 and 6.2 show the perceptions to be stationary and invertible.

    21

  • Claim 1 The perceptions implied by the use of an order up-to target Znt + nt k is of the following

    ARMA(n, 0) process for interpolations:

    (B) (Zt ) = at

    where (B) = 1Pni=1

    1nB

    i and the following ARMA(n, n) process for extrapolations:

    (B) (Zt ) = (B) at

    where (B) = 1Pni=1

    kn B

    i, and (B) = (B) .

    Lemma 6.1 The time series

    (B) (Zt ) = at

    where (B) = 1Pni=1

    1nB

    i is stationary and invertible.

    Lemma 6.2 The time series

    (B) (Zt ) = (B) at

    where (B) = 1Pni=1

    kn B

    i, and (B) = (B) is stationary and invertible.

    Note here that the mean of the demand process assumed in the perception is the same as

    actual since the moving average is used intuitively to estimate the actual level rather than assume

    a preconceived one. The explicit form of forecasting used for either interpolation or extrapolation

    here is the first explicit form, that is, J = 1. The perception implied by interpolations matches

    with how forecast errors are calculated for the moving average, thus note that it is an ARMA

    (n, 0) process. The perception implied by extrapolation matches with how inventory targets are

    set, thus note that the moving average dimension of this perception calculates the simple average

    of the previous n forecast errors and then combines it with the safety factor k. For completeness

    we provide the results for inventory levels and orders that are associated with the perceptions in

    Claim 1, given that demand follows the ARMA process (1).

    Proposition 6.3 The use of the moving average of the n previous demand realizations for an order

    up-to Znt + nt k inventory policy is associated with a time series of inventory It which is ARMA

    p, qIwith the form

    (B) It = I (B) aIt

    and:

    22

  • aIt = KIat q

    I = 2n+ q

    KI = 1 I (B) = (B) (B) (B) 2

    and associated with a time series of orders Ot which is ARMA

    p, qO

    with the form

    (B) (Ot ) = O (B) aOt

    and:aOt = K

    Oat qO = 2n+ q

    KO = 1 + 1+kn O (B) =

    1

    BKO 2(B)(1B)

    BKO+ (B)(B)(1B)

    BKO

    (B)

    6.2 Exponential Smoothing

    Exponential smoothing creates forecasts Ft which are a weighted average between the previous

    periods forecast and the current demand realization, that is,

    Ft = Zt + (1 )Ft1, (15)

    where 0 < < 1. We assume that the forecasts are combined with a safety stock T for an order

    up-to target.

    Muth (1960) shows that the exponential smoothing is optimal for an ARIMA(0,1,1) process.

    Thus the use of exponential smoothing can be considered to assume this non-stationary perception

    (see Section 7). However the exponential smoothing approach was designed before the optimality

    result by Muth was found. Furthermore the use of the exponential smoothing approach reflects

    less of an intended application to non-stationary process since there are specific approaches based

    on exponential smoothing designed to deal with trends and seasonality, see Holt (1957), Winters

    (1960), and Holt (2004). We propose therefore a stationary perception associated with the use of

    the exponential smoothing approach which can be determined as follows. Expanding (15) above

    we get,

    Ft = Zt + (1 ) (Zt1 + (1 )Ft1)

    = Xi=1

    (1 )i1 Zti+1

    mXi=1

    (1 )i1 Zti+1

    where the last approximation follows since there is some m for which (1 )m and higher powers

    can be considered negligible. This implies the following claim for consistent misaligned perceptions

    which categorize the use of exponential smoothing for setting inventory policies. Lemma 6.4 shows

    that the perception is stationary and invertible.

    23

  • Claim 2 The perception implied by using exponential smoothing with a weighting factor is the

    following ARMA(m) process for both interpolations and extrapolations:

    (B) (Zt ) = at

    where (B) = 1Pmi=1 (1 )

    i1Bi.

    Lemma 6.4 The time series

    (B) (Zt ) = at

    where (B) = 1Pmi=1 (1 )

    i1Bi is stationary and invertible.

    Note here that the mean of the demand process assumed in the perception is the same as the

    actual mean, that is = , as should be expected since exponential smoothing intuitively does

    not incorporate a preconceived mean. Here the explicit form used could be either of the three

    explicit forms since the perceptions implied by interpolations and extrapolations are the same. For

    completeness we provide the results for inventory levels and orders that are associated with the

    perceptions in Claim 2 and are corollaries of Theorems 4.2 and 4.3:

    Proposition 6.5 The use of exponential smoothing with weighting factor is associated with a

    time series of inventory It which is ARMAp, qI

    with the form

    (B) (It T ) = I (B) aIt

    and:aIt = K

    Iat qI = m+ q

    KI = 1 I (B) = 1KI

    (B) (B)

    and associated with a time series of orders Ot which is ARMA

    p, qO

    with the form

    (B) (Ot ) = O (B) aOt

    and:aOt = K

    Oat qO = m+ q

    KO = 1 + O (B) =(B)(1B)

    BKO

    (B)

    6.3 Discussion

    The above examination reveals a couple of interesting points about our perceptions framework.

    Firstly, from our examination of the use of the moving average forecasting, we see that it is very

    plausible for a setting where the perceptions implied by the extrapolation and interpolation are

    24

  • different. The setting also suggests reasons why the perceptions could be different as here, extrap-

    olations seem directly related to the form of the inventory policy leaving interpolations to be more

    focused on estimating demand shocks. Secondly from our examination of exponential smoothing,

    there might be more than one perception which fits a particular extrapolation and interpolation

    combination, as the exponential smoothing is optimal for a non-stationary demand process. How-

    ever we claim that its use suggests a stationary perception which we provide and show the expected

    orders and inventory as a result of this perception.

    7 Perceptions Framework for Suboptimal policies: ConsistentMis-aligned Non-stationary Perceptions

    In this section we complete the presentation of our perceptions framework with a treatment of

    demand modeled as non-stationary demand processes. For brevity, we will only consider consistent

    misaligned perception. As in Section 4, by characterizing the stochastic processes for the inventory

    levels and order levels given the same misaligned perception in both interpolations and extrapola-

    tions, we establish this perception as the basis for the link between the process characteristics of the

    system. After the results provided below, we describe the more interesting relationships between

    the categorizing perception and the process results. This misaligned perception is the following

    ARIMAp, d, q

    process:

    (B) (Zt ) = (B) at (16)

    where (B) = (B)d and the implied ARMA(p, q) process on the dth difference,

    (B)d (Zt )

    = (B) at, is stationary and invertible.

    We describe the approaches for extrapolation and interpolation which are analogous to their

    descriptions in section 4. Consider extrapolations, that is our one period ahead forecasts Zet , from

    origin t. As with stationary misaligned perceptions, Zet can be defined in the following three ways

    given the three explicit forms of the forecasts for an ARIMA process:

    Zet (1) = +1 (B)

    B(Zt ) +

    (B) 1B

    at (17)

    or

    Zet (2) = + (B) 1

    Bat (18)

    where (B) = 1 +Pj=1 jB

    j = (B) / (B) or

    Zet (3) = +1 (B)

    B(Zt ) (19)

    25

  • where 1Pj=1 jB

    j = (B) = (B) / (B) . Note that in the first two explicit forms of Zet we use

    the random shock terms at to reflect interpolations with misaligned perceptions. Also note that the

    third form does not use any interpolations. Also recall that for an ARIMA time series Zt, although

    dZt is stationary and invertible, Zt is not stationary though it is invertible. Therefore the infinite

    sum (B) above, is not summable and so does not converge in any sense. Alternatively we can

    rewrite (18) in terms of the dth difference which does have a converging sum of random shocks. As

    a notational device, we will refer to this infinite random shock form to represent the forecast based

    on infinite random shock form of the dth difference.

    Consider now the perceptions for interpolations. Recall that it is the difference between the

    actual realization of the dth difference and the one period ahead forecast of the dth difference that is

    used to generate the realization for the demand errors at. We can again let Zit denote the one period

    ahead forecast of demand used to create the one period ahead forecast of the dth difference and

    denote three forms for Zit as Zit (1), Z

    it (2) and Z

    it (3) which again coincide sequentially with the three

    explicit forms listed in Section 2.1. Again as with Proposition 4.1, we can write at = (B) (Zt )

    where (B) = (B) / (B) for either of the three forms Zit (j) . Theorems 7.1 and 7.2 below

    characterize the stochastic processes for the inventory levels and orders.

    Theorem 7.1 A misaligned implied perception is associated with a time series of inventory It

    which is ARMApI , dI , qI

    with the form

    I (B)It T I

    = I (B) aIt

    and:aIt = K

    Iat dI = max(0, d d) I (B) = (B) (B)max(0,dd)

    KI = 1 qI = p+max(0, d d) + q T I = T +1B(B)

    B

    ( )

    pI = q + p I (B) = 1KI( (B) (B))

    This time series is independent of the explicit forecasting form used for either interpolations or

    extrapolations and the dIth difference of inventory time series It is stationary. The time series is

    also invertible if the demand process is invertible.

    Theorem 7.2 A misaligned perception attributed to both the interpolations and the extrapolation

    is associated with a time series of orders Ot which is ARMApO, d, qO

    with the form

    O (B) (Ot ) = O (B) aOt

    and:

    26

  • aOt = KOat

    O (B) = (B) (B)

    KO = 1 + 1 1 qO = maxp+ q + d, q + q 1

    pO = q + p O (B) = (B)(B)

    KO+ (B)

    ((B)(B))BKO

    This time series is independent of the explicit form used for either interpolations or extrapola-

    tions and the dth difference of the order time series Ot is stationary but not necessarily invertible.

    Perceptions Perspective on the Inventory Level Process: Theorem 7.1 shows that

    for consistent misaligned perceptions, the inventory level will follow a non-stationary stochastic

    process if the perception represents a lower difference than that of the demand, irrespective of

    accuracy of the autoregressive and moving average operators of the perception. These operators

    of the perception still determine the autoregressive and moving average operators of the stochastic

    process for the inventory level.

    Perceptions Perspective on the Order Process: Theorem 7.2 shows that for consistent

    misaligned perceptions, orders will exhibit the same number of differences as demand. The order

    of the moving average operator for the order process is increasing in the number of differences in

    the perception d provided d is large enough.

    8 Conclusion

    In this paper we proposed a perceptions framework for categorizing a range of inventory decision

    making settings in a single-stage supply chain. The perceptions framework is based on forecasting

    with Auto-Regressive Integrated Moving Average (ARIMA) time series models. In order to cate-

    gorize these settings, we make the strong assumption that such settings can be grouped based on

    the combination of inventory levels, orders and demand persistently exhibited within the system,

    despite these settings being otherwise considered different from each other. These settings are then

    categorized by conceptual perceptions of demand which underpin an analytic link between inventory

    levels, orders and actual demand, a link that we establish in the paper. Our perceptions perspective

    relies on both interpolations and extrapolations using these ARIMA models. Optimal inventory

    policies can be categorized as perceptions that are aligned with reality and deviations from such

    polices as perceptions that are not aligned. With respect to misaligned perceptions, there are two

    variations. Consistent misaligned perceptions have the same perception for both interpolations and

    extrapolations while inconsistent misaligned perceptions do not. In support of our framework we

    categorize commonly used forecasting approaches using our perceptions perspective.

    27

  • For stationary processes we examined the inventory level process and the order process re-

    sulting from misaligned perceptions. In order to examine these effects, we separately examined

    autoregressive based misalignments and moving average based misalignments. Any misalignment

    in perception increases the variance of inventory level process over that of aligned perceptions.

    Monotonic properties related to change in parameters of misaligned perception depended on actual

    demand. For autoregressive based misalignment and moving average based misalignment, there

    was some support for variance of inventory levels increasing with difference in coefficients of oper-

    ators between perception and actual demand. For moving average based misalignment, monotonic

    properties for the variance of orders were found for changing the parameters of actual demand and

    keeping the misaligned perception constant, in particular, variance increased with an increase in

    the coefficients of the moving average operator of demand. For autoregressive based misalignments,

    variance of orders increased with an increase in the coefficients of the autoregressive operator of

    the perception.

    We expect our proposed perceptions framework will be useful for more robust examinations of

    supply chain management dynamics. In particular we believe we help broaden the language that can

    be used surrounding inventory decision making. So called optimal policies are a very restricted class

    of inventory dynamics both analytically and conceptually. A categorization, such as ours, which

    incorporates some latitude in describing and modeling inventory policies independent of costs,

    creates the opportunity for addressing a wider range of real-world situations where optimality is

    not an objective, either because costs are not known, are subjectively perceived or the goals of the

    organization are construed and pursued differently.

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    9 Appendix:

    9.1 Proofs of Theorems 3.1, 3.2, 4.2, 4.3 and Proposition 3.3.

    In this appendix we provide selected proofs and only parts of said proof so as to give the reader a

    sense of the mechanics used to get our results. These mechanics characterize all of the proofs for the

    results in this manuscript. All proofs can be found in an online supplement at

    http://www.people.hbs.edu/nwatson/.

    Proof of Theorem 3.1

    By definition we have

    It = T + Zt1 Zt.

    Recall

    Zt1 = +1 (B)

    B(Zt1 ) +

    (B) 1B

    at1.

    We also have that

    Zt = +1 (B)

    B(Zt1 ) + (B) at.

    This gives

    It T = (B) 1

    Bat1 (B) at.

    = at.

    30

  • Proof of Theorem 3.2

    Recall

    Zt = +1 (B)

    B(Zt ) +

    (B) 1B

    at.

    Now Ot = Zt + Zt Zt1 which gives

    Ot = Zt +1 (B)

    B(1B) (Zt ) +

    (B) 1B

    (1B) at.

    Since (Zt ) = ( (B) / (B)) at we get

    (B) (Ot ) = (B) at +1 (B)B +B (B)

    B (B) at +

    ( (B) 1) (1B)B

    (B) at

    =B + 1 (B)B +B (B)

    B (B) at +

    ( (B) 1) (1B)B

    (B) at

    =

    (B)

    B (B) (B) (1B)

    B+

    (B) ( (B) 1) (1B)B

    at

    =

    (B) (B)

    B+ (B)

    at.

    The above order process can be rewritten

    (B)d (Ot ) = (B) (B)

    B+ (B)

    at,

    with autoregressive order p, difference order d and moving average order

    max {p+ d, q 1} . Note first term in polynomial (B)(B)B + (B) is 1+1 1 so the above can

    be normalized as

    (B)d (Ot ) = (B) (B)

    BKO+

    (B)

    KO

    aOt ,

    where KO = 1 + 1 1 and aOt = KOat.

    Proof of Proposition 3.3

    We have

    V ar (Zt ) = V ar (B)

    (B)at

    = 2

    1 +

    Xi=1

    2i

    !while

    V ar (Ot ) = V ar (B) (B) (B)BKO

    + (B)

    (B)KOaOt

    = V ar

    (B)

    (B)BKO 1 +BBKO

    aOt

    = 2

    KO

    21 +

    Xi=1

    2i

    !

    31

  • From algebra we have that KO = 1 + 1 and KOi = i+1 for i 1. This implies that

    V ar (Zt ) < V ar (Ot ) 1 = 1 1 > 0.

    Proof of Theorem 4.2

    We only show the proof for extrapolating with the second explicit form of forecasting, i.e.,

    Zet1 (2) . From the definition of the inventory policy, we have

    It = T + Zet1 (2) Zt.

    Since

    Zet1 (2) = + (B) 1

    Bat1,

    and we also have that

    Zt = + (B) at,

    this gives

    It = T + (B) 1

    Bat1 (B) at ( ) .

    Since at = (B) (Zt ), (Zt ) = (B) at, (B) = (B) / (B) and (B) = (B) / (B) , if

    we let again =1B(B)

    B

    ( ) , we get our claim for J = 2:

    It T

    =

    (B) 1B

    (B) (Zt1 ) (B) at ( )

    =1 (B)

    B(Zt1 ) +

    1 (B)

    B

    ( )

    (B) (B) at ( )It T

    =1 (B)

    B (B) at1 (B) at

    = (B) (B) at

    (B) (B)It T

    = (B) (B) at.

    For all forms J = 1, 2, 3 the process for It is: (B) (B)It T

    1B(B)

    B

    ( )

    =

    (B) (B) at where the order of (B) (B) is p+ q while the order of (B) (B) is p+ q. Note

    that if both demand and the perception are assumed stationary and invertible, It is also stationary

    and invertible. Note first term in polynomial (B) (B) is 1 1 1

    which we can use to

    normalize the process as in Theorem 3.2.

    Proof of Theorem 4.3

    We only show the proof for J = 3. Consider Zet (3),

    Zet (3) = +1 (B)

    B(Zt ) .

    32

  • Now Ot = Zt + Zet (3) Zet1 (3) which gives

    Ot = Zt +1 (B)

    B(1B) (Zt )

    Ot = Zt +1 (B)

    B(1B) (Zt )

    =1 (B) (1B)

    B(Zt )

    +

    1 (B)

    B(1B)

    ( )

    Since1(B)B (1B)

    ( ) = 0, (Zt ) = ( (B) / (B)) at and (B) = (B) / (B) we get

    Ot =1 (B) / (B) (1B)

    B( (B) / (B)) at

    (B) (B)Ot

    =

    (B) (B) (1B)

    B

    ! (B) at.

    For all forms J = 1, 2, 3, the process for Ot is the same, that is, (B) (B)Ot

    =

    (B)(B)(1B)B

    (B) at where the order of (B) (B) is p+q while the order of

    (B)(B)(1B)

    B

    (B)

    is max {p+ q, q + q 1} . Note that if both demand and the perception are assumed stationary

    and invertible, Ot is also stationary but not necessarily invertible. Note first term in polynomial(B)(B)(1B)

    B

    (B) is 1 + 1 1.

    33